Let be $X:\Omega\to\mathbb{N}$ a Poisson-distributed random variable that denotes the occurences of events during a time interval $[0,t]$, where $t>0$.
This is a classic setting in our stochastic lecture when talking about the Poisson distribution. However, I can't find anything about the sample space $\Omega$? Our professor circumvents/avoids this issue every time.
How do we define/construct an appropriate sample space $\Omega$ if we deal with Poisson-distributed random variables? Maybe at this stage of my studies this is a bit too advanced. Or is there any chance to construct a sample space the same way we did when dealing with other discrete random variables, like coin flips, balls in bins etc...?
(There is a related question What is the probability space for Poisson process? but it didn't fully answered my problem)